What is the area of the smallest portion of the circle?
Two perpendicular chords divide a circle with radius of 13 inches into four parts. If the perpendicular distances of both chords are 5 cm each from the center of the circle. Find the area of the smallest part.
The area of a quarter of the circle is 169pi/4. From this, take away two circular sectors which are each subtended by the angle arctan(5/12). The formula for the area of such sectors is (1/2)r^2theta, so each of these sectors has area (169/2)arctan(5/12). The area we now have is
169pi/4 – 169arctan(5/12).
It remains to take away two triangles, each with sides 13, 7, and 5sqrt(2). By Heron’s formula, these triangles have area 35/2. Taking away two of these, we are left with the answer
169pi/4 – 169arctan(5/12) – 35 = 31.0126.
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The area of a quarter of the circle is 169pi/4. From this, take away two circular sectors which are each subtended by the angle arctan(5/12). The formula for the area of such sectors is (1/2)r^2theta, so each of these sectors has area (169/2)arctan(5/12). The area we now have is
169pi/4 – 169arctan(5/12).
It remains to take away two triangles, each with sides 13, 7, and 5sqrt(2). By Heron’s formula, these triangles have area 35/2. Taking away two of these, we are left with the answer
169pi/4 – 169arctan(5/12) – 35 = 31.0126.
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